3-Factor-criticality of vertex-transitive graphs

TL;DR

This paper characterizes when connected vertex-transitive graphs of odd order are 3-factor-critical, showing they are so if and only if they are not cycles, extending understanding of matching properties.

Contribution

It provides a complete characterization of 3-factor-criticality in simple connected vertex-transitive graphs of odd order, identifying cycles as the exception.

Findings

Connected vertex-transitive graphs of odd order ≥ 5 are 3-factor-critical iff not cycles.

All such graphs of odd order are factor-critical.

Cycles of odd order are the only exceptions not 3-factor-critical.

Abstract

A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-Factor-critical graphs and 2-factor-critical graphs are factor-critical graphs and bicritical graphs, respectively. It is well known that every connected vertex-transitive graph of odd order is factor-critical and every connected non-bipartite vertex-transitive graph of even order is bicritical. In this paper, we show that a simple connected vertex-transitive graph of odd order at least 5 is 3-factor-critical if and only if it is not a cycle.